The Mathematics of Game Shows

Bowen Kerins

Research Scientist, EDC (and part-time game consultant)

[email protected] Overview Game shows are filled with math problems… • Contestants • How do I play best? • How much is enough? • Producers • How do I build a fun game to watch? • How will contestants behave? • How much money are we giving out? PRIZES!

Want to win?

We’ll need some volunteers for games.

You may leave here with fabulous prizes!

(Warning: definition of fabulous may vary.) Personal Encounters February 2000: Millionaire (episode #49)

(for $1000: How many degrees in a right angle?) Personal Encounters February 2000: Millionaire (episode #49)

(I got the next question wrong.) Personal Encounters April 2004: The Price Is Right

(Double overbid on the showcase! Bummer.) Personal Encounters July 2007: National Bingo Night

(I also worked on “Show Me The Money” and “The Singing Bee”… which, four years later, is still on the air.) Personal Encounters Since Then…

Recent work: “in-development” shows for Endemol, Gurinco, and Ryan Seacrest Productions. (He’s older than me.) Expected Value Hour

$.01 $1,000 $1 $5,000 $5 $10,000 $10 $25,000 $25 Better known as $50,000 $50 $75,000 $75 Deal or No Deal $100,000 $100 $200,000 $200 $300,000 $300 $400,000 $400 $500,000 $500 $750,000 $750 $1,000,000 Expected Value Hour $.01 The “fair deal”: $1,000 $1 $5,000 $5 $10,000 $10 Multiply each $25,000 $25 outcome by its $50,000 $50 probability… $75,000 $75 $100,000 $100 $200,000 $200 Total: $410,210 $300,000 $300 $400,000 $400 $500,000 $500 $750,000 $750 Fair deal: ~$102,500 $1,000,000 Expected Value Hour $.01 The “bank offer”: $1,000 $1 $5,000 $5 $10,000 $10 Guarantee, almost $25,000 $25 always less than fair $50,000 $50 value $75,000 $75 $100,000 $100 $200,000 $200 Fair deal: ~$102,500 $300,000 $300 $400,000 $400 Offer: $82,000 $500,000 $500 $750,000 $750 Deal or No Deal? $1,000,000 Expected Value Hour

$.01 What’s the expected $1,000 $1 value of the initial $5,000 $5 $10,000 $10 board? $25,000 $25 $50,000 $50 How does it compare $75,000 $75 $100,000 $100 to the first offer? $200,000 $200 $300,000 $300 How does it compare $400,000 $400 to how much money $500,000 $500 players actually win? $750,000 $750 $1,000,000 Expected Value Hour

$.01 $1,000 $1 Initial board… $5,000 $5 Fair deal: $131,477 $10,000 $10 $25,000 $25 $50,000 $50 First offer: $75,000 $75 $100,000 $100 ~$8,000-$20,000 $200,000 $200 $300,000 $300 $400,000 $400 The first offers are $500,000 $500 terrible! Why? $750,000 $750 $1,000,000 Expected Value Hour

$.01 $1,000 $1 Actual average $5,000 $5 winnings per player: $10,000 $10 $25,000 $25 $122,500 $50,000 $50 $75,000 $75 Initial board’s $100,000 $100 expected value: $200,000 $200 $300,000 $300 $131,477 $400,000 $400 $500,000 $500 $750,000 $750 (Close! Why the difference?) $1,000,000 Expected Value Hour

$.01 $1,000 $1 Offer percentages $5,000 $5 (compared to fair $10,000 $10 value, by round): $25,000 $25 $50,000 $50 11%, 21%, 36%, $75,000 $75 50%, 62%, 73%, $100,000 $100 88%, 92%, 98% $200,000 $200 $300,000 $300 $400,000 $400 $500,000 $500 (Commercial break…) $750,000 $750 $1,000,000 Sponsored by… CME Project • Four-year, NSF-funded curriculum written by EDC • Published in 2008 by Pearson Education • 25,000+ students use it nationally: Boston, Chicago, Pittsburgh, Des Moines… and more Fundamental Organizing Principle The widespread utility and efectiveness of mathematics come not just from mastering speciﬁc skills, topics, and techniques, but more importantly, from developing the ways of thinking—the habits of mind—used to create the results. CME Project Overview By focusing on habits of mind… • Coherent curriculum, fewer chapters • Closely aligned to Common Core’s Standards of Mathematical Practice (several ideas come from CME) • Closely aligned to NCTM’s Reasoning and Sense- Making goals (several examples come from CME) • General-purpose tools help students get the big ideas

Summer sessions in New England! June 27-29, July 18-20, August 1-5 (we also do house calls… but now, back to the show) The Price Is Right

• Now in its 39th year • Lots of good math problems! • Also a huge sample size of repeatedly- played games (for agonizing detail, visit http://tpirsummaries.8m.com)

Who wants to play?? Four Price Tags Place a price next to each item. If it’s the right price, you win the prize!

Slumdog Millionaire

Glee: The Game

Word Wear

Jenga Four Price Tags Place a price next to each item. If it’s the right price, you win the prize!

Slumdog Millionaire Good luck… you’ll need it. $9.99 Glee: The Game $10.00 Word Wear $10.09 $10.59 Jenga Audience, any advice? Four Price Tags So, how did you do…?

Slumdog Millionaire $10.00

Glee: The Game $10.59

Word Wear $9.99

Jenga $10.09 The Producers’ Question

If I keep offering this game repeatedly, how many prizes will I give away, on average?

Answer by calculating the expected value for the number of prizes per game.

There are 24 different ways the player can place the price tags. (Why?) 24 Ain’t That Many Here are the 24 ways to place the price tags:

ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA Gotta Score ‘Em All For each of the 24 ways, count the number of price tags placed correctly.

ABCD 4 BACD 2 CABD 1 DABC 0 ABDC 2 BADC 0 CADB 0 DACB 1 ACBD 2 BCAD 1 CBAD 2 DBAC 1 ACDB 1 BCDA 0 CBDA 1 DBCA 2 ADBC 1 BDAC 0 CDAB 0 DCAB 0 ADCB 2 BDCA 1 CDBA 0 DCBA 0 The expected value is… This frequency chart shows the number of ways to get each result.

# prizes # ways 4 1 2 6 1 8 0 9 TOTAL 24 The expected value is… This frequency chart shows the number of ways to get each result.

# prizes # ways The total number of prizes is 4 1 4 x 1 + 2 x 6 + 1 x 8 2 6 1 8 … 24 prizes and 24 ways. Divide to 0 9 find… TOTAL 24 Hey, it’s 1! A Second Opinion Reconsider the problem from how Sue sees it. (Sue Sylvester, from Glee.)

ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA A Second Opinion Light up all the places where B (the Glee game) is correctly placed.

ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA A Second Opinion Glee is won one-fourth (6 out of 24) of the time… and all the prizes are like that.

ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA It’s always 1! With four prizes, each is won 1/4 of the time.

4 x (1/4) = 1

With five prizes, each is won 1/5 of the time. With n prizes, each is won 1/n of the time.

n x (1/n) = 1

Clever methods can “beat” enumeration. Two Extensions 1. As the number of prizes grows, what happens to the probability of winning nothing at all?

2. The mean (average) number of prizes given away is always 1. What happens to the standard deviation?

These two problems have great conclusions, which this slide is too small to contain… (We’ll be right back.) Sponsored by… Rice-A-Roni • It’s “The San Francisco Treat”! • A favorite since 1958 • 25,000+ people eat it nationally: Boston, Chicago, Pittsburgh, Des Moines… and more

(And we’re back.) The #1 Game on TPIR is…

PLINKO! The #1 Game on TPIR is…

PLINKO! Plinko is played so often that great data is available: 2000-2011: played 308 times! Total chips: 1,227 Total chips in $10,000 space: 176 (14.3%) Total winnings: $2,214,600 ($1,805 per chip) Average winnings per play: $7,190 Backtracking Plinko

How much is this Plinko chip worth right now? Backtracking Plinko 100

500

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row… row… the last We know that spot. at of a chip is the value entry Each Backtracking Plinko 100 300

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below it! of thetwo the is number each up… the bottom Work from mean mean Backtracking Plinko 100 300 300

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below it! of thetwo the is number each up… the bottom Work from mean mean Backtracking Plinko 100 300 300 413

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2750

1688

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below it! of thetwo the is number each up… the bottom Work from mean mean Backtracking Plinko 100 300 413 300 413

500 494 525

750 575

1000 1131 625

1688 500

2750 2781

5000 3875

10000 5000 3875

5000 3875

2750 2781

1688

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1000 1131 625

750 575

525 494 500

300 413

413 300 100

below it! of thetwo the is number each up… the bottom Work from mean mean Backtracking Plinko 100 300 413 300 453 413

500 494 525

750 813 575

1000 1131 625

1956 1688 500

2750 2781

5000 3328 3875

10000 5000 3875

5000 3875 3328

2750 2781

1956 1688

500

1000 1131 625

750 813 575

525 494 500

300 453 413

413 300 100

below it! of thetwo the is number each up… the bottom Work from mean mean Backtracking Plinko 100 453 300 413 300 453 413

500 633 494 525

750 813 575

1000 1384 1131 625

1956 1688 500

2642 2750 2781

5000 3328 3875

10000 3328 5000 3875

5000 3875 3328

2642 2750 2781

1956 1688

500

1384 1000 1131 625

750 813 575

633 525 494 500

300 453 413

413 453 300 100

run… run… a longisn’t But this out. evens run, it all In thelong Backtracking Plinko 100 453 300 413 300 453 413 543

500 633 494 525

1009 750 813 575

1000 1384 1131 625

1956 1688 2013 500

2642 2750 2781

5000 3328 3875 2985

10000 3328 5000 3875

5000 2985 3875 3328

2642 2750 2781

1956 1688

2013 500

1384 1000 1131 625

1009 750 813 575

633 525 494 500

300 453 413 543

413 453 300 100

run… run… a longisn’t But this out. evens run, it all In thelong Backtracking Plinko 100 543 453 300 413 300 453 413 543

500 776 633 494 525

1009 750 813 575

1000 1511 1384 1131 625

1956 1688 2013 500

2499 2642 2750 2781

5000 3328 3875 2985

10000 2985 3328 5000 3875

5000 2985 3875 3328

2499 2642 2750 2781

1956 1688

2013 500

1384 1000 1511 1131 625

1009 750 813 575

633 525 494 776 500

300 453 413 543

413 453 543 300 100

run… run… a longisn’t But this out. evens run, it all In thelong Backtracking Plinko 100 543 453 300 413 300 453 413 659 543

500 776 633 494 525

1143 1009 750 813 575

1000 1511 1384 1131 625

1956 1688 2005 2013 500

2499 2642 2750 2781

2742

5000 3328 3875 2985

10000 2985 3328 5000 3875

2742 5000 2985 3875 3328

2499 2642 2750 2781

1956 1688 2005

2013 500

1384 1000 1511 1131 625

1009 1143 750 813 575

633 525 494 776 500

300 659 453 413 543

413 453 543 300 100

run… run… a longisn’t But this out. evens run, it all In thelong Backtracking Plinko 100 543 659 453 300 413 300 453 413 659 543

500 776 901 633 494 525

1143 1009 750 813 575

1000 1511 1574 1384 1131 625

1956 1688 2005 2013 500

2374 2499 2642 2750 2781

2742

5000 3328 3875 2985

10000 2742 2985 3328 5000 3875

2742 5000 2985 3875 3328

2374 2499 2642 2750 2781

1956 1688 2005

2013 500

1384 1000 1574 1511 1131 625

1009 1143 750 813 575

633 525 494 901 776 500

300 659 453 413 543

659 413 453 543 300 100

run… run… a longisn’t But this out. evens run, it all In thelong Backtracking Plinko 100 543 659 453 300 413 300 453 413 659 780 543

500 776 901 633 494 525

1143 1238 1009 750 813 575

1000 1511 1574 1384 1131 625

1956 1688 2005 1974 2013 500

2374 2499 2642 2750 2781

2742

2558 5000 3328 3875 2985

10000 2742 2985 3328 5000 3875

2742 2558 5000 2985 3875 3328

2374 2499 2642 2750 2781

1956 1688 2005 1974

2013 500

1384 1000 1574 1511 1131 625

1009 1143 1238 750 813 575

633 525 494 901 776 500

300 659 780 453 413 543

659 413 453 543 300 100

each slot! each from chip dropping for value expected we findthe At the top a Backtracking Plinko 100 543 659 780 453 300 413 300 453 413 659 780 543

1009 500 776 901 633 494 525

1143 1238 1009 750 813 575

1000 1511 1574 1384 1606 1131 625

1956 1688 2005 1974 2013 500

2374 2499 2266 2642 2750 2781

2742

2558 5000 3328 3875 2985

10000 2742 2985 2558 3328 5000 3875

2742 2558 5000 2985 3875 3328

2374 2499 2266 2642 2750 2781

1956 1688 2005 1974

2013 500

1384 1000 1574 1511 1606 1131 625

1009 1143 1238 750 813 575

1009 633 525 494 901 776 500

300 659 780 453 413 543

659 413 453 543 780 300 100

each slot! each from chip dropping for value expected we findthe At the top a Good Plinko Advice

Where you drop Plinko chips matters a lot!

Drop Above Chip EV $10,000 $0 $1,000 $500 $100 Good Plinko Advice

Where you drop Plinko chips matters a lot!

Drop Above Chip EV $10,000 $2,558 $0 $2,266 $1,000 $1,606 $500 $1,009 $100 $780 Good Plinko Advice

Where you drop Plinko chips matters a lot!

(Did they build the board Drop Above Chip EV this way on purpose?) $10,000 $2,558 $0 $2,266 $1,000 $1,606 $500 $1,009 $100 $780 Good Plinko Advice

Where you drop Plinko chips matters a lot! If you ever get on… Drop Above Chip EV DROP IT IN $10,000 $2,558 $0 $2,266 THE MIDDLE!!! $1,000 $1,606 Actual average winnings: $500 $1,009 $1,805 per chip $100 $780 ($753 lost per chip… 1,227 times) Sponsored by… Gold Bond Medicated Powder • Developed in 1882 by pharmacists in Rhode Island • Gold Bond: Does what it says. • It’s got triple action, whatever that means!

(Let’s get back to the games already…) The #2 Game on TPIR is…

ANY NUMBER! We don’t need a Plinko board to play this one.

Who wants to play?? CHEESE 3 2 0 7

BUS 6 9 5

PIGGY BANK 1 8 4 Any Number

Assuming the player is just picking randomly (which seems about right), what is the probability that they win the big prize?

This is a hard question! Any Number Assuming the player is just picking randomly, what is the probability that they win the big prize?

This question would be a lot easier if the big prize had 3 digits instead of 4…

The probability of winning the big prize must be less than 1/3. Solving by Simulation There are 10! = 3,628,800 different ways the player can pick numbers. 10! is a much bigger number than 24, so enumerating by hand is impractical. One option is to simulate running the game a large number of times. Here’s 10,000 trials:

Cheese: 2,605 (26.05%) Bus: 3,682 Piggy Bank: 3,713 Solving by Tree Diagram Solve a simpler version! If there’s one number in each prize left, there is a 1/3 chance of winning the big prize. If there’s two numbers left in the big prize and one in each of the small prizes, there is a 2/4 x 1/3 = 1/6 chance of winning the big prize. This is like a coordinate system: P(2,1,1) = 1/6. Continue and “build out” until you find the answer at P(4,3,3). (Build a 3-D model!) Solving by Enumeration For computers, 3.6 million isn’t that big, it’s around the number of 5-card poker hands. A computer can try all 10! ways the game could be played: Cheese: 933,120 (25.71% = 9/35) Bus: 1,347,840 (37.14% = 13/35) Piggy Bank: 1,347,840 (37.14% = 13/35)

This is different from what the simulation found; its probability is an estimate. Solving by Being Clever Let’s play a different game called Any Number But That One. You pick a number; if it’s in a prize, that prize explodes. You win the last prize standing. Say you pick 3… cheese explodes! (Oops.)

What’s the probability of the cheese standing alone? Solving by Being Clever This game doesn’t last nearly as long… On the first pick, 6 of the 10 numbers explode one of the two small prizes. (Which is good.) After you blow up one small prize, the next pick is decisive: 3 of the 7 numbers explode the other small prize, and then it’s all cheese. 6/10 x 3/7 = 9/35 (25.71%) Hey, it’s the same probability… Solving by Being Clever Imagine being forced, before playing Any Number, to write down all 10 digits in the order you plan to call them. Here’s how it matches up with Any Number But That One: • The first exploding number you pick is the final digit that you never plan to pick. • The second exploding number is the last one in the other prize you didn’t complete. In long games, it is often easier to look at what doesn’t happen instead of what does. Historical Data Players win Any Number more often than predicted by chance. 2000-2011 257 plays Big prize (A New Car!): 92 (35.80%) Small prize: 91 Piggy Bank: 74

Players often guess the first digit of the car. Also, 0 and 5 are more likely to appear there. (Our last commercial break…) Classroom Interlude In my teaching, I found some game shows worked better than others. Mostly I used games for test review, but also for openers or wrap-ups.

Good Bad Press Your Luck Jeopardy! (yes, bad) Card Sharks Deal or No Deal Millionaire Twenty-One High Rollers Newlywed Game Sponsored by… the Mathematical Practices Institute • EDC’s new professional development program • Curriculum-neutral, focused on Common Core’s eight Standards for Mathematical Practice • One-day and one-week seminars available

Visit the MPI website and blog: www.edc.org/cme/mpi That website again is: www.edc.org/cme/mpi A Quick Game… Who’s got a #2 pencil? Stand up. Now roll this die number cube, but before you do, pick one: • Try to roll a 1 through 5 to win $2, or • Try to roll a 6 to win $20.

You must pick in advance! What’s it gonna be??? The Same Game…? Let’s raise the stakes… hypothetically. Now roll this die number cube, but before you do, pick one: • Try to roll a 1 through 5 to win $25,000, or • Try to roll a 6 to win $250,000.

Why is this so different? How do players behave? It’s National Bingo Night! NBN was The Price is Right with bingo instead of shopping. (It only lasted six episodes. Wonder why…) Its games were interesting probability problems, but usually no strategy. Except for Bingo 365… Who wants to play? Get Out Your Bingo Cards… The final game of the show is Bingo 365. If you complete a bingo before the contestant wins, you win all remaining prizes! (Ties will be broken by a math problem.) Important: if you are one number away from a bingo, STAND UP so we can tell you are one away. Bingo 365

For each bingo ball, the contestant guesses whether the next ball will be higher or lower than the one that just came out. If they are right, the ball number is added to their score. 75 is better than 23. The first ball doesn’t score any points. Bingo 365

The contestant wins if they get a total of 365 points or more before anyone in the audience completes a bingo. TV Show audience: 200 players, one card each. This audience: 100 players, two cards each. I swear this was actually on network TV. Bingo 365

42 Bingo balls go from 1 to 75.

Is the next ball higher or lower than 42? Bingo 365

42 60

Is the next ball higher or lower than 60? Bingo 365

42 60 5

Is the next ball higher or lower than 5? Bingo 365

42 60 5 30 Remember: if you are one away, STAND UP. Is the next ball higher or lower than 30? Bingo 365

42 60 5 30 46

Is the next ball higher or lower than 46? Bingo 365

42 60 5 30 46 24

Is the next ball higher or lower than 24? Bingo 365

42 60 5 30 46 24 1

Is the next ball higher or lower than 1?

(Don’t think too hard about this one.) Bingo 365

42 60 5 30 Remember: if you are one away, STAND 46 24 1 47 UP. Is the next ball higher or lower than 47? Bingo 365

42 60 5 30 46 24 1 47

64 Is the next ball higher or lower than 64? Bingo 365

42 60 5 30 46 24 1 47

64 39 Is the next ball higher or lower than 39?

(What would you do here? This one is kind of in the middle.) Bingo 365

42 60 5 30 46 24 1 47

64 39 72 Is the next ball higher or lower than 72? Bingo 365

42 60 5 30 Getting closer…? 46 24 1 47

64 39 72 56 Is the next ball higher or lower than 56? Bingo 365

42 60 5 30 46 24 1 47

64 39 72 56 Is the next ball higher 67 or lower than 67? Bingo 365

42 60 5 30 46 24 1 47

64 39 72 56 Is the next ball higher 67 41 or lower than 41? Bingo 365

42 60 5 30 46 24 1 47

64 39 72 56 Is the next ball higher 67 41 18 or lower than 18? Bingo 365

42 60 5 30 Bad time for such a low number! 46 24 1 47

64 39 72 56 Is the next ball higher 67 41 18 3 or lower than 3? Bingo 365

42 60 5 30 46 24 1 47

64 39 72 56 Is the next ball higher 67 41 18 3 or lower than 48? 48 Bingo 365

42 60 5 30 46 24 1 47

64 39 72 56 Is the next ball higher 67 41 18 3 or lower than 71? 48 71 Bingo 365

42 60 5 30 46 24 1 47

64 39 72 56 (At NCTM 2011, the 67 41 18 3 contestant won with this ball.) 48 71 26 Bingo 365 Strategy Did strategy change as the game went on? One strategy is to go higher if in the bottom half, otherwise lower. Is this the best strategy? Or what? What’s this got to do with summations? Balance? Similar triangles?? Players did not follow good strategy on the actual show: the first contestant to play Bingo 365 picked “lower” on a 30. (He lost.) More to Explore Many related topics are asked about in CME Project Precalculus, and in the Park City Math Institute materials at www.mathforum.org/pcmi/hstp/sum2007/morning • How can spinners or dice be represented by polynomials? • How would expected value change if the Plinko board were really, really tall? • Would it be reasonable for 30 of 100 people to win Any Number by chance? 35? 40? Thanks and good luck!

Any questions?

Bowen Kerins Research Scientist, EDC (and part-time game consultant)

[email protected] www.edc.org/cme/mpi